MATHEMATICAL PROGRAMMING APPROACH TO COURSE-TEACHING ASSISTANT ASSIGNMENT PROBLEM. Fadime Üney-Yüksektepe 1, İlayda Karabulut 1

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1 MATHEMATICAL PROGRAMMING APPROACH TO COURSE-TEACHING ASSISTANT ASSIGNMENT Fadime Üney-Yüksektepe 1, İlayda Karabulut 1 1 İstanbul Kültür University, Faculty of Engineering and Architecture Industrial Engineering Department, İstanbul, Turkey f.yuksektepe@iku.edu.tr, i.karabulut@iku.edu.tr Abstracts: In this study, assignments of teaching assistants to problem session of the courses are investigated. In general, the departments determine the course timetable at the beginning of each semester. For some of the courses there exist problem sessions that should be assist by the teaching assistant of the department. Due to the capacity limit of the computer laboratories and classrooms, a problem session of a course should be divided into two or more different groups. Furthermore, as teaching assistant are graduate students and have to take some graduate courses, their unavailable times should be considered while planning the assignments. On the other hand, in order to be fair the total workload of each teaching assistant has to be balanced. Moreover, learning effect depending on the previous year assigned courses, preliminary work, other grading related works and number of assigned courses are also considered while modeling the problem. A mixed-integer linear programming model is developed in order to solve the courseteaching assistant assignment problem. The developed model is tested on the Spring 2011 semester data of a private university. Hence, the proposed model is coded in GAMS and solved using CPLEX 13.0 solver. Optimal solution of the problem is obtained in a very short amount of time. When the optimal solution of the proposed model is compared with the current plan of the department, the fairness and the effectiveness of the proposed result is observed. Due to the simplicity and the efficiency of the proposed model, its applicability and the satisfaction of the teaching assistants will increase. Keywords: Assignment Problem, Workload Balance, Mathematical Modeling Introduction In universities, after the course timetable is scheduled, the teaching assistants are assigned to the problem sections that are assisted by the teaching assistants of the department. Some difficulties occur in the existing situation while assigning the teaching assistants in the problem sections. The main and the most important problem is the unfair distribution of the work load between the teaching assistants. This problem brings up huge dissatisfaction between teaching assistants. Besides, a waste of time takes place to come with the common idea by the longstanding meetings. Therefore, disagreement and injustices appear in the assignment of the teaching assistants to the problem sections. In addition, same teaching assistants are assigned in the same problem sections which lead to improve oneself in the same field. Thereby, teaching assistants improve themselves just in a specific field that is undesirable. The main factor behind this study is based on the need of improvement the assignment of the teaching assistants to the problem sections. In this study, assigning the Istanbul Kultur University (IKU) Industrial Engineering Department's teaching assistants to the problem sections is handled. Due to the capacity limit of the computer laboratories and classrooms, a problem session of a course needs to be divided into two or more different groups. Also, teaching assistants are graduate students and have to take some graduate courses. Therefore, their unavailable times is considered while planning the assignments. On the other hand, in order to be fair the total workload of each teaching assistant has to be balanced. IKU, Industrial Engineering Department has seven teaching assistants, ten courses with two or more problem sections and three courses without problem sections which requires teaching assistants. Furthermore, each course has different preparation time for its 878

2 problem section. Moreover, each problem section has some responsibilities which are taken by teaching assistants such as, attendance, quizzes, homeworks, projects, etc. In this problem, we proposed a mixed integer programming model in order to solve the course-teaching assistant assignment problem. The model is tested by the data obtained from Department of Industrial Engineering at IKU. Therefore, the developed model is solved for a real life problem. The remaining part of this paper is organized as follows. In the next section, proposed mathematical model is described in detail. Data related to the considered course-teaching assistant assignment problem and the results for the numerical tests are given in Section 3. Paper concludes with conclusion and suggestions in Section 4. Proposed Mathematical Model The mixed-integer linear programming (MILP) model presented in this section considers the assignment problem between the problem sections and the teaching assistants which intends the effective available schedule. The developed model is tested on the Spring 2011 semester program of IKU. Each day is divided into 10 time periods and current course timetable has 50 time periods. Hence, the following indices are used to develop the proposed model: i represents the teaching assistants (1,, I) and j imposes the courses (1,, J). In addition, k is used for sections (1,, K), and t is used for time periods (1,, 50). In order to solve this problem, the following sets are assumed to be known: SC jk : Set of course-section pairs j-k WPSC j : Set of courses j with problem sections TC j : Set of courses j that should need only one teaching assistant P ijk : Set of preassigned teaching assistant i to section k of course j m it : Set of teaching assistant i that are not available at time period t The following parameters are assumed to be known: S jkt :1 if section k of course j is scheduled at time t; 0 otherwise Pre j : Amount of weekly preparation time needed to assist course j ow j : Amount of weekly other work (quizzes, homeworks, projects, etc.) time needed to assist course j le ij : 1 if course j will be given by teaching assistant i for the first time; 0.8 otherwise The following decision variables are necessary to model the problem: Y ijk :1 if section k of course j is assigned to teaching assistant i; 0 otherwise Z ij :1 if teaching assistant i is assigned to course j; 0 otherwise wmax: maximum workload wmin: minimum workload tw i : total working amount of teaching assistant i The following mixed integer programming model is developed to solve course-teaching assistant assignment problem: min z wmax - wmin (1) subject to 1, (2) = 1, (3) = 0, (4) 879

3 =, (5) 2, (6) 2, (7) = 1, (8), (9), (10) = 1, (11), (12),, (13) {0,1}, (14) {0,1}, (15) 0, (16) 0, 0 (17) The objective function (1) is to minimize the difference between total workloads of teaching assistants. Constraint (2) ensures that at least 1 course j must be assigned to each assistant i. Constraint (3) shows that if the assistant i is preassigned to a course j, the assistant must be assigned to that course. If assistant i is not available for the defined time interval due to her/his graduate courses, the assistant must not be assigned to the course that is scheduled for that time interval. This restriction is given by constraint (4). Moreover, constraint (5) is necessary to ensure that the sum of the total problem section time, preparation time, homeworks, and quizzes, and so on is equal to the total workload for each assistant. Constraint (6) represents that assistant i must be assigned at most two problem sectioned-courses. Course j must be assigned to a maximum of two assistants i. This restriction is given by constraint (7). In addition, constraint (8) represents that course j-section k pairs must be assigned to an assistant i. Constraint (9) shows that the total time spent during the semester must be less than the maximum workload. On the other hand, constraint (10) indicates that the total time spent during the semester must be greater than the minimum workload. Only one assistant should be assigned to the courses j that is specified as is shown in constraint (11). The relationship between the two binary variables is given in constraints (12) and (13). Finally, constraints (14) and (15) give the integrality of the decision variables and constraints (16) and (17) give the non-negativity of the decision variables. Computational Tests In this section, computational results of the developed MILP model that is tested on a real data are presented. Before analyzing the results, some information related to the real data is given. The developed model is tested on the Spring 2011 semester courses of a private university. Academic semester is handled as 14 weeks. According to the academic program of the courses, daily 10-hour time intervals are determined. Moreover, student numbers are taken from the current semester for each course. The time for the problem sections, evaluation time for quizzes, homeworks and projects and etc. are some of the responsibilities of teaching assistants. Therefore, the approximate time for these responsibilities is taken from each teaching assistant in the current semester and the results are used as numerical data. The student numbers in each section, total problem section hours, total preparation hours for the each course and other grading related works such as quizzes, homeworks and projects and etc. are given in Table 1. As teaching assistants are graduate students and have to take some graduate courses, their unavailable time is considered while planning the assignments. Also, there are some courses which have one hour practice section in different times. Therefore, these courses should be given with one teaching assistant. According to some technical training, some courses need to have technical knowledge and skills. For this reason, some assistants are preassigned to the specific courses. In addition, learning effects are discussed in the model. Namely, if course j is given by teaching assistant i for the first time, then the work load will be same which 880

4 numerical data was taken from the teaching assistants. If teaching assistant i is assigned to the course j in the previous years, the work load will decrease by 20%. The reason behind that is the preparation time for the course in the second time will be less than the first time as the teaching assistants will have necessary documents for the preparation in the second year. This is referred as the learning effect. Although, preparation time will decrease by 20%, other grading related works such as quizzes, homeworks and projects and etc. will remain same. Table 1. Course-section pairs with workloads and student numbers. TOTAL TOTAL # of PS PRE STUDENTS (HOURS) (HOURS) TOTAL OW (HOURS) IE 002 Work Study IE 011 Enterprise Resource Planning IE 202 Algor. and Intro. to Program, IE 202 Algor. and Intro. to Program, IE 202 Algor. and Intro. to Program, IE 202 Algor. and Intro. to Program, ,74 IE 250 Intro. to Industrial Engineering IE 404 Numerical Analysis, IE 404 Numerical Analysis, IE 404 Numerical Analysis, IE 411 Applied Statistics, IE 411 Applied Statistics, IE 411 Applied Statistics, IE 421 Operations Research I, IE 421 Operations Research I, IE 421 Operations Research I, IE 421 Operations Research I, IE 463 Engineering Economics, IE 463 Engineering Economics, IE 612 Engineering Experimental Design, IE 612 Engineering Experimental Design, IE 612 Engineering Experimental Design, IE 631 Management Informations Systems, IE 631 Management Informations Systems, IE 631 Management Informations Systems, IE 652 Production Planning and Control, IE 652 Production Planning and Control, IE 854 Quality Engineering IE 890 Final Project Based on the data obtained from the teaching assistants, proposed model is formulated in GAMS 23.6 [1] and solved by using CPLEX 12.0 [2] solver in order to obtain the optimal results. The runs were executed on 881

5 a computer which has a 1.66 GHz processor and 2 GB of RAM. Optimal solution of the problem is obtained in a very short amount of time. The characteristics of the developed model for this real data are given in Table 2. Table 2. Characteristics of the proposed model. ITEM VALUE # of Constraints 427 # of Binary Variables 294 # of Continuous Variables 10 # of Iterations Solver Memory (MB) 4 MB CPU Time (seconds) TEACHING ASSISTANS Table 3. Comparison of the current situation and proposed model. CURRENT SITUATION PROPOSED SOLUTION TOTAL WORKLOAD (hour) WITHOUT TOTAL WORKLOAD (hour) WITHOUT TA TA TA TA TA TA TA Max Min Average Std. Deviation Objective Function When the optimal solution of the proposed model is compared with the current plan of the department, the fairness and the effectiveness of the proposed result is observed. As it is shown in Table 3, the objective function which describes the difference between the workloads of teaching assistants is decreased from hours to 4.5 hours. That means that the proposed model's results give a positive contribution about the distribution of workload. Moreover, it is clearly shown that the standard deviation between the workloads of teaching assistants is also decreased from hours to hours. The proposed model provides the highest level of job satisfaction between teaching assistants. For instance, in the current situation TA 1 had hours of workload in that semester on the other hand; in the proposed solution TA 1's workload has decreased to hours to balance the workload between the teaching assistants. Besides, in the current situation, TA 5 had hours of workload in that semester. In contrast, in the proposed solution TA 5's workload has increased to hours to stabilize the workload between the teaching assistants. Also, it is evident that on average the workload between the teaching assistants is approximately 170 hours. Due to the 882

6 simplicity and the efficiency of the proposed model, its applicability and the satisfaction of the teaching assistants will increase. Conclusions In this paper, the assignments of teaching assistants to problem session of the courses are analyzed. A mixed integer programming model is developed to solve this problem optimally to have a contribution in the Industrial Engineering Department of Istanbul Kultur University. In the problem, 7 assistants are considered to balance workloads between each other. Thus, each assistant will have approximately equal workloads by the proposed solution. By the help of developed model, unlike the current situation, teaching assistants are assigned to the courses which does not has any practice sections. Therefore, it will certain that each course will have its own teaching assistant. In addition, teaching assistants are assigned to the different courses than the previous semesters that they can develop themselves in different areas. As total workload was based on data which was taken from the teaching assistants, the maximum preparation time for each course is determined. When results are evaluated, in the current situation the distribution of workload was unbalanced. However, in the proposed solution, assignments of workloads are determined in a balanced way. Furthermore, the model will be applied in coming periods, thus it will help to prevent the loss of time during the meetings for assignment. It is easy to understand the developed model which gives a good solution as well as increase the satisfaction of teaching assistants. Possible future work could be to include the distribution of administrative tasks' workload and a comprehensive plan according to these tasks. Also, a user friendly decision support system can be generated. Acknowledgement We are thankful for the support of the head of Department of Industrial Engineering at Istanbul Kültür University through the completion of this research. References 1. Brooke, A., Kendrick, D., Meeraus, A., Raman, R., GAMS:A User's Guide. GAMS Development Co., Washington, DC. 2. Ilog, CPLEX 12.0 User's Manual, ILOG S. A. See website 883